Minimax Optimality and Spectral Routing for Majority-Vote Ensembles under Markov Dependence

Tree ensembles such as random forests (RFs) and gradient boosting machines (GBMs) are among the most widely used supervised learners, yet their theoretical properties remain incompletely understood. We adopt a spectral perspective on these algorithms, with two main contributions. First, we derive minimax-optimal convergence for RF regression, showing that, under mild regularity conditions on tree growth, the eigenvalue decay of the induced kernel operator governs the statistical rate. Second, we exploit this spectral viewpoint to develop compression schemes for tree ensembles. For RFs, leading eigenfunctions of the kernel operator capture the dominant predictive directions; for GBMs, leading singular vectors of the smoother matrix play an analogous role. Learning nonlinear maps for these spectral representations yields distilled models that are orders of magnitude smaller than the originals while maintaining competitive predictive performance. Our methods compare favorably to state of the art algorithms for forest pruning and rule extraction, with applications to resource constrained computing.

We study the minimax estimation of the pseudo-spectral gap $γ_{ps}$ of a Markov chain from the observation of a single trajectory. Extending earlier works on finite-state space chains by Wolfer and Kontorovich (2024), we provide an estimator in the case of a compact, continuous state-space. Our estimator appears to be minimax-optimal on the class of chains with a transition operator with Lipschitz density, up to logarithmic terms. Our estimator, combined with Paulin's version of Bernstein inequality, allows to derive empirical concentration inequalities for Markov chains. This unlocks many applications, from confidence intervals, to active learning and generalization bounds. In particular, for rested bandits with Markov rewards, we provide a version of UCB with a regret bound similar to the one in the iid case, without assuming any knowledge on the pseudo-spectral gap of the chains.

Margin bounds explain why weighted votes generalize after training error vanishes, but dependent observations require concentration arguments that keep the dependence model and failure budget explicit. We study stationary mixing streams without treating dependence as a formal substitution of an "effective sample size." For a finite base class, we combine independent-block Rademacher theory for stationary $β$-mixing processes with a clipped margin loss to obtain an explicit uniform voting-margin certificate, whose dependence penalty is set by the number of approximately independent blocks and the coupling budget $4(μ-1)β(a)$. A black-box transfer theorem shows that any i.i.d. voting-margin guarantee remains valid on a regularly spaced subsequence, with only an additive $β$-mixing failure probability. For votes fixed independently of a stationary finite-state reversible Markov stream, spectral concentration gives the cleaner $\sqrt{t_{rel}/n}$ rate, and a sticky-refresh construction shows this exponent is unavoidable. The constant is statistic-specific: the effective sample size is $n/τ_{int}(g)$, with $τ_{int}(g) \le 2t_{rel}-1$. When the dependence scale is unknown, a batch-means Student-t interval is valid up to an explicit remainder vanishing in the number of batches, with width governed by $τ_{int}(g)$ rather than the worst-case $t_{rel}$. Experiments show: the learned-vote certificate is nonvacuous once the block count is large; a two-timescale construction gives a batch-means interval 8.1x narrower than the worst-case interval at nominal coverage (0.945); block-bootstrap and batch-means intervals undercover under strong dependence while the certificate stays valid; and on real, ordered UCI HAR sensor data the dependence-agnostic certificate reaches 0.95 coverage where naive intervals reach only 0.38.